If it is submitted by mail, the Trademark Office will use the date of receipt of the documents as the filing date. If the Trademark Office receives B's application documents first, then the application date is earlier, and the trademark will be granted to B. . If the documents from B and A are received on the same day, the applicant will be required to submit proof of the date of first use of the trademark within thirty days. As long as it can be proven that A used the trademark before B, the trademark right will be granted to A.